Intermediate systems and higher-order differential constraints Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2018, 11(5), 550—560

YflK 519.21

Intermediate Systems and Higher-Order Differential Constraints

Oleg V. Kaptsov*

*

Institute of Computational Modelling SD RAS Academgorodok, 50/44, Krasnoyarsk, 660036

Russia

Received 10.03.2018, received in revised form 20.04.2018, accepted 06.06.2018

A method for constructing solutions of nonlinear partial differential equations with two independent variables is proposed. The method is based on the search for so-called intermediate systems, each solution of which satisfies the initial equation. The main attention is paid to a second order nonlinear wave equation. We give examples of intermediate systems and corresponding solutions.

Keywords: differential constraints, defining equations, invariant manifolds. DOI: 10.17516/1997-1397-2018-11-5-550-560.

Introduction

One of the first methods of integrating nonlinear partial differential equations, proposed by Monge and Ampere, can be formulated as follows [1,2]. Suppose we are given an equation of the second order

depending on an arbitrary constant c € R such that any non-singular solution satisfies (1). The equation (2) is called the first or intermediate integral of the second order equation (1). Further development of this method was obtained in the works of Darboux and his followers. The key point in the Darboux method is that it is necessary to find an equation of order n that depends on a constant and is in involution with (1). The details of the Monge-Ampere and Darboux methods are well described in [3]. Modern aspects of these methods and some generalizations are presented in [4,5]. It should be noted that the equations (1), to which the Monge-Ampere and Darboux methods are applicable, are exceptional. For example, among the equations of the form

only the Liouville equation is integrated by the Darboux method. On the other hand, the Goursat problem of describing all equations of the form

integrable by the Darboux method remains open [6].

In this paper, we propose an approach to the integration of equations of the type (1), based on the finding of such systems

§(t, X, U, Ut, Ux, Utt, Utx, Uxx) = 0.

It is necessary to find an equation of the first order

V(t, x, u, ut, ux) + c = 0,

(1)

(2)

Utx = f (u)

(3)

Utx = F (t,X,U,Ut ,Ux)

ut = G(x, u, ui,..., um),

d¿ u

* kaptsov@icm.krasn.ru © Siberian Federal University. All rights reserved

H(x, u, ui,..., un) = 0, (5)

that each solution of this system satisfies the equation (1) as well. We suppose the manifold (5) is invariant under the equation (4) [7] and say that the system (4), (5) is intermediate.

The article has the following structure. In Section 2 we construct intermediate systems for equations of the type (3) using their higher symmetries. The higher symmetries [8] are also called generalized or Lie-Backlund symmetries [9,10]. It is shown that if the equation (3) admits an operator of higher symmetry

ft ^ ft «i + £D"-(n)-k -

n=i

where Dx is the operator of total derivative with respect to x, then the system

ut + V = 0, Dx(n)+ f = 0 (6)

is intermediate for the equation (3). To solve the system (6), we use the Runge-Kutta methods for the approximate solutions of ordinary differential equations. Some solutions of the sine-Gordon equation are given. In the third section, intermediate systems for the equation (3) are found without the use of symmetries. It turns out that the equation

ut = eu(u2 - ul)

has a simultaneous solution with the sinh-Gordon equation, and the third order evolution equation

ut = eu(u^ — 2u^u2)

has a simultaneous solution with the Tzitzeica equation.

The fourth section is devoted to the construction of invariant manifolds for diffusion equations

ut = (ukux)x. (7)

For special exponents k we found differential constraints depending on the third, fourth and fifth derivatives with respect to x. Examples of exact solutions of the equation (7) obtained by integrating differential constraints are given.

1. Intermediate systems generated by symmetries

Among the nonlinear equations of the form

utx = f(u), (8)

as is known [11], only equations with right hand sides f = ai exp(u) + a2 exp(—u), f = ai sin(u), f = ai exp(u) + a2 exp(—2u) (ai, a2 G R) have higher symmetries. We recall that the operator

X = nd-u + E Wn) dUTn

n=l

is a higher symmetry of the equation (8) if the equality

DD df

DtDxV =

du

is satisfied according to the equation (8) and its differential consequences. Here and thereafter, Dt, Dx will mean the operators of total derivatives with respect to t and x respectively [9].

To each operator X one can associate the differential equation

n = 0. (9)

This leads to the system (8), (9) for the function u. In this section we consider examples of such systems. We begin with the Tzitzeica equation

utx — exp(u) — exp(—2u) = 0. (10)

It has a denumerable set of higher symmetries [11]. Let us investigate the compatibility of a system consisting of (10) and the equation

u5 + 5(u2u3 — u-u3 — ui u2) + uf =0. (11)

Here and thereafter, we will use the notation ui = . The left-hand side of (11) defines a fifth order higher symmetry of the Tzitzeica equation (10). From the system (10), (11) you can get more convenient one. Indeed, differentiating (11) with respect to t and substituting the mixed derivatives, according to (11), we obtain a new equation

uiu3(e3u +4) + 3u2(-e3u + 1) + 2u1 u2(2e3u - 1) - u1(e3u + 1)

u4--e^2--()

Further, differentiating the last equation with respect to t and substituting the mixed derivatives from (11), we obtain the evolution equation

ut + (2e-2" - e")(u3 + 2u№) =Q 2uiu3 — (u2 — ul)2

By direct calculations one can verify that the equation (12) is an invariant manifold of the evolution equation (13).

We recall the notion of the invariant manifold [7]. Let us consider an evolution equation

ut + F(t, x, u, ui,..., un) = 0 (14)

and an ordinary differential equation

h = um + G(x, u, ui,..., um—i) = 0. (15)

The equations (14) and (15) form the manifolds in the jet space [8].

Definition 1. The manifold (15) is called invariant under the equation (14) if the relation

Dth = 0 (16)

is satisfied due to (14), (15) and their differential consequences with respect to the variable x. In this case, the system (14), (15) will be called passive (in the narrow sense).

Remark. The general definition of a passive system in the analytic case is given in [12].

It should also be noted that the Tzitzeica equation is a differential consequence of the system (10), (11). More precisely, the following formula holds

^ , n (2e—2u — eu)(u2 + u2)2 , N ,

Dx (ei) + —---e2 = utx — exp(u) — exp(—2u),

(2uiu3 — (u2 — u-)2)2

where ei is the left-hand side of (13), e2 is the left-hand side of (12). Hence, any solution of the system (12), (13) is a solution of the Tzitzeica equation.

If we set four initial conditions

diu

d— (to,xo) = ci, i = 0,..., 3, ci e R,

then from [5] it follows that the system (12) and (13) will have a unique local solution. It is assumed that the initial data are not special, i.e. denominators in (12) and (13) are not equal to zero. We say that the solution of the system depends on four constants in the neighborhood of a nonsingular point.

As a second example, consider the sine-Gordon equation

utx — sin(u) = 0, (17)

supplemented by an ordinary differential equation of the fifth order

5 3

U5 + ^(uU + uiu2) + -8uf = 0, 2 8

generated by one of the higher symmetries. Repeating the arguments given above for the Tzitzeica equation, from the last two equations we obtain the passive system

e? = u4 + (12u2u1 + 8u?ui tg(u) — 4u2 + 3u1 tg(u))/8 = 0,

2u3 + u?

e4 = ut +4CQS(u)8u?ui — 4u2 + 3u1 =0.

The solution of this system also depends on four constants, and the sin-Gordon equation is a differential consequence of this system

8cos(u)(4u2 + u4) .

^x(e4) + (8u? ui — 4u2 + 3u4)2 e? = utx — Sln(u).

Definition 2. A passive system

G = ut + F (t, x,u,ui,... ,uk) = 0, H (t, x, u, ui,..., um) = 0

is called intermediate for equation

utx + f (x,u,ui,... ,un) = 0, (18)

if

Dx(G)+ rH = utx + f, where r is a function of t,x,u,ui,... ,up (p <k).

We have given above two examples of intermediate systems. Now we will present one of the ways of constructing such systems.

Proposition 1. Let the operator

d ~ d

Y = F (x,u,ui, ...,um) — + J2 Dn(F) dun, (19)

n=i n

dF

where —-=0 is the higher symmetry of the equation (18). Then the system

dum

ut + F = 0, Dx(F) — f = 0 (20)

is intermediate for the equation (18).

Proof. The equality obviously holds

utx + f = Dx(ut + F) — (Dx(F) — f). (21)

The equation (18) is invariant under a translation of t. As is well known, the symmetries of the equation form a Lie algebra [8,10]. Hence, according to the condition of the Proposition 1, the operator

d ~ d

X = (ut + F) du + £ Dn(ut + F) dun

n=i

is also a symmetry of the equation (18). Obviously, the operator X is a symmetry of the equation Dx(ut + F) = 0 and this equation is invariant under the traslation. Hence, it admits an operator Y. As shown in [5], this means that the system (20) is passive. □

As an example, consider the sine-Gordon equation (17). The operators of higher symmetries (19) for this equation form an infinite-dimensional Lie algebra L [10, 11]. In this case, the function F is represented as a linear combination (with real coefficients) of the functions Fi = ui, Fi+i = Li(Fi), where L = D^ + u2 — uiD—iu2 is the so-called recurrence operator. In particular, the functions F2 and F3 are given by formulas

1 5 3 F2 = u3 + - u3, F3 = u5 + -(u2u3 + u-u2) + - u-.

2 2 8

As repeatedly noted, the Lie algebra L is also the algebra of symmetries of the mKdV equation

13

ut = u3 + ^ ui.

Proposition 1 clarifies the connection between the solutions of higher order mKdV equations and the solutions of the sine-Gordon equation. The higher order mKdV equations are the evolution equations

ut + F = 0,

where the function F is a linear combination of the functions Fi = ui,...,Fi+i = Li(Fi). According to Proposition 1, the system

ut + F = 0, Dx(F) + sin(u) = 0 (22)

is passive and each of its solutions satisfies the sine-Gordon equation. In particular, the 1-soliton solution of the sine-Gordon equation

u = 4 arctg(exp(mx + t/m + n)), m,n G R

satisfies the system (22), where F = c0F3 + ciF2 + c2Fi, and the constants c0, ci, c2, m are related by the formula c0m6 + cim4 + c2m2 + 1 = 0.

In addition to the 1-soliton solution, the intermediate system (22) has other solutions. As shown in the monograph [7], the standard numerical Runge-Kutta methods can be used to solve systems of the type (22). If we introduce new functions ul = ui, ..., u4 = u4, then the system (22) with F = c0F3 + ciF2 + c2Fi is rewritten as two systems of ordinary differential equations of the first order in t and x respectively. Given five initial conditions, we solve the system of ordinary differential equations in t, and then use the obtained data to solve the system in terms of x. In this manner, we obtain a solution of the system (22) satisfying the initial data u(0,0) = 3.14, ui(0,0) = 2, u2(0, 0) = 0, u3(0,0) = —2, u4(0,0) = 0, u5(0,0) = 10 and c0 = —1, ci = c2 = 0. The graph of the solution is shown in Fig. 1.

Fig. 1

2. Intermediate systems of nonlinear wave equations

In this section we construct examples of intermediate systems for the nonlinear wave equation

utx + f (u) = 0, (23)

without using the symmetries. In the general case, the problem of finding intermediate systems is very difficult. It requires the solution of nonlinear partial differential equations with several independent variables.

We will look for intermediate systems in the form of polynomials with respect to ui (i > 0) with coefficients depending on u. By analogy with (20), we assume that the system has the form

ut = aiu2 + a2ui, (24)

h = Dx(aiu2 + a2ui) + f = 0, (25)

where ai, a2 are only functions of u. The first equations in the system is invariant under a scale transformation of independent variables.

Since the system must be passive, it is proved in [7] that the function h must satisfy the so-called linear defining equation of the form

Dth — r0D2x h — rDxh — rih = 0, (26)

where r0, r, ri are some functions of u,ui,... ,un and the relation (26) must be fulfilled because of the equation (24), i.e. the derivatives ut,utx,... are expressed in terms of aiu2 + a2ui, Dx(aiu2 + a2ui), and so on. The left hand side of the equation (26) is a polynomial with respect to ui,... u5, and all its coefficients must therefore be zeros. Collecting similar terms for u5 yields the relation a2 — r0ai = 0. Hence, the formula holds

r0 = a1.

Next, we collect the terms with u1u4 and find the function r

r = (ai + 2a2)ui.

If we collect similar terms containing u3, then it is easy to express the function ri

ri = 2u2 (a'i + a,2) + u'[(a2 a'i/ai + 3a2). Collecting of similar terms with u2 gives the relation

ai(ai + a2)f = 0. Since the case ai f = 0 is not of interest to us, the equality

a2 = —ai

must hold. Collecting terms with u2u2 and taking into account the previous equality, we obtain the equation

ai ai = (ai )

which has a solution

ai = c exp(ku), where c, k are arbitrary constants. This yields

a,2 = —ck exp(ku).

Substituting the found functions into the defining equation (26) and collecting terms with u-j2, we obtain the equation

f" — 4k2f = 0.

From the equation we find

f = ci exp(2ku) + c2 exp(—2ku), ci, c2 G R.

Thus, the intermediate system (24), (25) has the form

ut = ceku(u2 — ku2), h = Dx(ceku(u2 — ku2)) + cie2ku + c2e—2ku = 0,

and the corresponding equation (23) is

utx = cie2ku + c2e—2ku.

If c = k = 1, then the linear defining equation (26) is

Dth — eu(D2x h — uiDxh — 4u2h) = 0.

Now consider the third-order evolution equation

ut = F = aiu3 + a2uiu2 + a3u3, (27)

with ai,a2,a3 G R. The additional equation is given by the formula

h = DxF + f = 0. (28)

The system (27), (28) must be passive and, consequently, satisfy a linear defining equation of the form

Dth — sDx h — r0D2x h — rDxh — rih = 0, (29)

where s,r0,r, ri are functions of u,ui,u2,u3 to be determined together with h.

The left-hand side (29) is a polynomial in ui,... ,u7. All the coefficients of this polynomial must be zero. It is easy to calculate that the coefficient of u7 is equal to ai — sai. It follows that s = ai. Collecting terms with u6,u5,u4, we find the functions r0,r,ri:

ro = (ai + a2)ui, r = (a2 + 3a3)ui + 2a2u2,

ri = [u3ai(2a'i + a2) + uiu2 (aia + 2aia2 + 6aia3) + uf (aia + 4aia3 )] /ai.

The remaining terms on the left hand side (29) form a polynomial in three variables ui, u2, u3. Equating the coefficients of this polynomial to zero, we obtain an overdetermined system of ordinary differential equations for functions ai, a2, a3, f. This system is easy to solve. Omitting the computational details, we see that there are only three types of solutions. We present the final form of the equations (27) and the corresponding functions f:

(1) ut = eu(u3 — 2uiu2),

(2) ut = u3 — ui/2,

(3) ut = u3 + ui/2,

where ci,c2 are arbitrary constants. The second but the first one is probably new.

f = cie2u + C2e-u, f = cieu + C2e-u, f = ci sin(u) + c2 cos(u),

and third evolution equations are well known,

3. Diffusion equations with constraints

In this section, we will continue constructing passive systems using the defining equations. Let us consider a nonlinear diffusion equation

Vt = (vkVx)x, k £ R.

We replace u by vk and obtain an equation with the bilinear operator [13]

ut = uuxx + bu,x, b = 1/k. (30)

As in the previous sections, we supplement the equation (30) by an ordinary differential equation

h = 0,

the right-hand side of which satisfies the linear defining equation

Dth — r0D2x h — rDxh — rih = 0, (31)

where r0, r, ri are functions of u,ui,u2,... . We assume that h has the form

h = u3 + aiuiu2 + a2ui + a3ui + a4.

Here ai,a2,a3, a4 are only functions of u. The left-hand side of the equation (31) is a polynomial in ui,..., u5. Equating the coefficients of this polynomial containing u5,u4,u3 to zero, we have

r0 = u, r = (2b + 3)ui, ri = (2aiu — ai)ui + (6b + 4 — 2aiu)u2.

The remaining terms in the polynomial lead to an overdetermined system of ordinary differential equations containing unknown functions ai,... ,a4. Omitting the long but easy calculations, we give the final result.

Proposition 2. For any b G R the linear defining equation has two solutions

hi = U3, h-2 = U3 + 2bu\u2/u — bui/u2. (32)

Moreover, there are additional solutions of (31) for the following values of b:

(a) b = —1, h = U3 + cu\, h = U3 +

'1UU\U2 — u1 — ClUl 2c2 — U2 :

(b) b = -2, h = «3 - ^ +2^1 + Clu2 + 2^,

u u2 u2

(c) b = -2/3, h = u3 + c1u-4/3 + c2,

/ 7\ 7 I/O A I U1 (U2 +2C1)

(d) b = -1/2, h = U3 + u(c2ui/2 + 2), where c,c1 ,c2 are arbitrary constants.

Remark. Some solutions of the equation (30) are given in [5,13,14]. In particular, for b = -1/2 this equation can be linearized.

As an example, we show how to find exact solutions of the system

2 2uuiu2 - u1 - c1«1

Ut = ««2 - u1, «3 +----2- = U. (33)

2c2 - u2

We rewrite the second equation in the form

«3(2c2 - u2)/«1 + 2u«2 - u2 = C1 and differentiate with respect to x. This leads to the fourth order differential equation

u4u1 - u2u3 = U

without any constants! Dividing the equation by u1u3 and integrating, we obtain

u3 + cu1 = U.

If c = -k2, then the general solution of the last equation is

u = S1Ckx + S2C-kx + S3,

where k, s1,s2, s3 are functions of t. We substitute this solution in the second equation of the system (33) and find

_ c1 - 2c2k2 + k2s3

S1 = W72 .

Next, we substitute the function u into the first equation of the system (33). As a result, we have the system of equations

k' = U, s2 = k2S2S3, S2s'2 - 2(s2)2 = (c1k2 - 2c2k4)s2.

If the constant m = -c1k2 + 2c2k4 is positive, then

S2 =-^-F= , d,1,d2 £ R;

d1etVm +

otherwise, we obtain

1

S2

d\ sin(^—m) + d2 cos(t\f—m) - 558 -

The functions s^ s3 are expressed from the above formulas. Another solution of the defining equation (31) is

7 _ U , (b + l)u1u2 „

h = u3---+--= 0. (34)

ui u

Moreover, the system (30), (34) is passive for any b £ R. If we divide h by u2, then it is easy to find the first integral and then reduce it to the first-order equation

ui = qiu-b + q2,

where q1,q2 are functions of t. For some values of b, we can construct solutions of the last equation in elementary functions. These solutions are related to the invariance of the equation (30) under the translations in t and x.

We give an additional statement which is verified by direct calculations. Proposition 3. The equations (30) and h = 0 form a passive system with the following values of the constant b and the function h:

3u2u3 , (b + 2)uiu3 2u3 u2

V b £ R h = u4---\---\--2---;

ui u uf u

7 -, , U2U3

b = —1, h = U4--;

Ul

6U1U3 4u2 18u?u2 8u"

b = —2, h = u4-----1--2---;

u u u2 u3

2uu 4u2 2u"u2

b = —2, h = u4-----1--2— ;

u u u2

b = —2/3, h = u4 + 4uu3;

3u

22

h MO h , u"u3 , u2 u1u2 .

b = —1/2, h = u4 ^^ + 2u — -4ÜT;

3uiu3 u2

b = —1/2, h = u4 ^^ + 2u '

b = 1/3, h = u5 + --(3uu2u4 + 7u2u4);

3uu"

1

uu1

o o Q 0 0 A \

+ 6u3 u-ju + 4u2u + 10u2 u^u — 6u2 u^);

b = —1, h = u5 +---— (—u3 u2u4 — 3u4u" u2 — 8u-u2uiu2 +

b = —1/2, h = u5 +--(uiu4 — 15u2u3);

2u

b = 1/2, h = u5 +--2 (12uuiu4 + 10u3u2 u + 3u3u2 + uj^ui);

4u2

b =1/4, h = u5;

b = 1/2, h = u5 + —(5uiu4 + 5u2u3). 2u

The work was supported by the Russian Foundation for Basic Research (grant 17-01-00332-a).

References

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[3] M.E.Goursat, Recherches sur quelques equations aux derivees partielles du second order, Annales de la Faculte de Toulouse, 2 serie, t. 1, 1899.

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Промежуточные системы и дифференциальные связи высших порядков

Олег В. Капцов

Институт вычислительного моделирования СО РАН Академгородок, 50/44, Красноярск, 660036

Россия

В 'работе предложен метод построения решений нелинейных уравнений в частных производных с двумя независимыми переменными. Метод основан на поиске так называемых промежуточных систем, каждое решение которых удовлетворяет исходному уравнению. Основное внимание уделяется нелинейному волновому уравнению второго порядка. Приведены примеры промежуточных систем и соответствующих решений.

Ключевые слова: дифференциальные связи, определяющие уравнения, инвариантные многообразия.